Constant-Factor Approximation to Deadline TSP and Related Problems in (Almost) Quasi-Polytime
Bibliographic record
Abstract
We investigate a genre of vehicle-routing problems (VRPs), that we call max-reward VRPs, wherein nodes located in a metric space have associated rewards that depend on their visiting times, and we seek a path that earns maximum reward. A prominent problem in this genre is deadline TSP, where nodes have deadlines and we seek a path that visits all nodes by their deadlines and earns maximum reward. Our main result is a constant-factor approximation for deadline TSP running in time O(n^O(log(nΔ))) in metric spaces with integer distances at most Δ. This is the first improvement over the approximation factor of O(log n) due to Bansal et al. [N. Bansal et al., 2004] in over 15 years (but is achieved in super-polynomial time). Our result provides the first concrete indication that log n is unlikely to be a real inapproximability barrier for deadline TSP, and raises the exciting possibility that deadline TSP might admit a polytime constant-factor approximation. At a high level, we obtain our result by carefully guessing an appropriate sequence of O(log (nΔ)) nodes appearing on the optimal path, and finding suitable paths between any two consecutive guessed nodes. We argue that the problem of finding a path between two consecutive guessed nodes can be relaxed to an instance of a special case of deadline TSP called point-to-point (P2P) orienteering. Any approximation algorithm for P2P orienteering can then be utilized in conjunction with either a greedy approach, or an LP-rounding approach, to find a good set of paths overall between every pair of guessed nodes. While concatenating these paths does not immediately yield a feasible solution, we argue that it can be covered by a constant number of feasible solutions. Overall our result therefore provides a novel reduction showing that any α-approximation for P2P orienteering can be leveraged to obtain an O(α)-approximation for deadline TSP in O(n^O(log nΔ)) time. Our results extend to yield the same guarantees (in approximation ratio and running time) for a substantial generalization of deadline TSP, where the reward obtained by a client is given by an arbitrary non-increasing function (specified by a value oracle) of its visiting time. Finally, we discuss applications of our results to variants of deadline TSP, including settings where both end-nodes are specified, nodes have release dates, and orienteering with time windows.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.014 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.009 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".